Oracle Separations for the Quantum-Classical Polynomial Hierarchy

• URL: http://arxiv.org/abs/2410.19062v1
• Date: Thu, 24 Oct 2024 18:12:03 GMT
• Title: Oracle Separations for the Quantum-Classical Polynomial Hierarchy
• Authors: Avantika Agarwal, Shalev Ben-David,
• Abstract summary: We study the quantum-classical hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers. We give a new switching lemma for quantum algorithms which may be of independent interest.
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• Abstract: We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any $d$, if we apply a random restriction to a function $f$ with quantum query complexity $\mathrm{Q}(f)\le n^{1/3}$, the restricted function becomes exponentially close (in terms of $d$) to a depth-$d$ decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.

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